On estimates for the rate of convergence of power penalty methods

The problem under consideration is the nonlinear optimization problem min f(x) text{ subject to } x in D={xin Bbb{R}^{n} mid F(x)=0, G(x)leq 0}, where fcolon Bbb{R}^{n}rightarrow Bbb{R}, Fcolon Bbb{R}^{n}rightarrow Bbb{R}^{l} and GcolonBbb{R}^{n}rightarrow Bbb{R}^{m} are sufficiently smooth mappings. For the solution of the problem, the authors use the power penalty function varphi_{c} = f(x) +c psi(x), where cgeq 0 is a penalty coefficient and psi(x)= (rho (Psi(x)))^{p}, Psicolon Bbb{R}^{n}rightarrow Bbb{R}^{l}times Bbb{R}^{m}, Psi(x)= (F(x),(max{0, G_{1}(x)}, dots, max{0, G_{m}(x)})). They investigate the rate of convergence of the power penalty function method